10.2 The Inverse Transform Method
Result 10.2.1. To simulate \(X\) with distribution function \(F\): generate \(U\sim \) UNIF\((0,1)\) and return \(X = F^{-1}(U)\). For a discrete distribution taking values \(x_1<x_2<\cdots \) with probabilities \(p_1,p_2,\dots \), return the smallest \(x_k\) for which \(p_1+\cdots +p_k \geq U\).
Example 10.2.2. Give an algorithm to simulate an EXP\((\lambda )\) variable, where \(\lambda \) is the rate.
Solution. Here \(F(x) = 1-e^{-\lambda x}\) for \(x>0\). Solving \(u = 1-e^{-\lambda x}\), \[e^{-\lambda x} = 1-u \quad \Longrightarrow \quad x = -\frac {1}{\lambda }\log (1-u).\] So generate \(U\) and return \(X = -\lambda ^{-1}\log (1-U)\). Since \(1-U\) is also uniform on \((0,1)\), the simpler \(X=-\lambda ^{-1}\log U\) has the same distribution and is what is normally used — one subtraction saved per observation, which matters when millions are drawn.
Note. The method needs \(F^{-1}\) in closed form, which is why it suits the exponential, the Weibull and the Cauchy, and does not suit the normal — whose distribution function has no elementary inverse. That gap is what the next two methods fill.
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